Ted Wong (Biology)
The Long Journey of a Simple Biological Model
Prepared by Anne Dalke
Additions, revisions, extensions are encouraged in the Forum
Ted invited us to "look under the hood" with him to see how a model "ticks," that is, how computational researchers like himself come up with math to represent "something that happens in the world." He began his demonstration of a thirty-year-old simple model by inviting us to consider what trade-offs happen in the course of constructing an equation for how plants might best allocate energy to maximize reproduction. The mathematical model he was using, which was designed by Dan Cohen in 1967(?), began with the presumption that a plant can either invest in reproductive structures (seeds) or "everything else" (vegetative structures such as roots, leaves and stems). The question motivating the construction of the model was how much a plant should allocate to each sort of investment, on each day, in order to maximize fitness. This is "the big question" in botany: what schedule of allotment of energy is likely to result in the biggest amount of seeds? How does a plant--and how might a plant best-- allocate its resources? Ted walked us through the process whereby that question was translated into math, producing a model for how the size of a vegetative body changes. It resulted in a "bang/bang" allocation schedule: the prediction that a plant will be most fecund if it allocates all of its resources to vegetative size in the early part of the season, and then switches to allocate all of its resources to reproductive growth. (The applicability of this model to our own calculations about when we should have our children were evoked, then quickly squelched.)
During the discussion that followed, it was asked how closely the model fit the empirical data (answer: lots of annuals don't do this, although certain weeds, and some crop plants which produce all their fruit @ the end of the season, do). We considered together a number of the "unrealistic assumptions" which underlay the construction of the model. Many of the decisions made in constructing the model were "not innocuous," either because some information was not accessible, or in order to keep the model mathematically doable. Perhaps the constants in the equation are not actually constants? Maybe the efficiency of the conversion to carbon dioxide changes? Maybe how much a plant can grow is not a simple proportionality function? Perhaps there are limits of scale, with larger structures being less efficient, because they need more energy for structural support? The model appeared to be founded on the presumption that fitness is best measured by large seed mass, but how those seeds are dispersed, and the importance of timing--when they come out in the course of the season--are variables that need to be taken into question. It was suggested that whether the model describes fitness is not an assumption of, or a problem with, the model (which technically is simply a model of reproductive biomass), but a question rather of how it is used, how it relates to the outside world, what it MEANS.
The answer to the main question we were asking--what allotment schedule will produce the highest fitness?--lies outside the model. But measuring plants to see if they follow the model's prediction may well only reinforce the paradigm it constructs. This is a metaquestion, scientific rather than mathematical in nature (i.e. a question about how we want to apply the model, rather than a technical question about the tool itself). Yet there also seems to be an interesting overlap between the two: certain aspects of the model are presumed because it is mathemathical. Is it at least entertainable that it is not possible to get math to act like the real world? That features of the tool make it impossible to do so?
A mini-industry of ecology papers were generated by Cohen's model. Dozens of new models have been published during the past fifteen years, each with a different elaboration or refinement of what Cohen had done: making constants into functions; adding other variables (such as "storage") to the two simple alternatives of vegetative and reproductive investment; considering how to account for the fact that a plant may diminish in size with time, as pieces die and fall off. Since Cohen's model is usuable only if the plant knows when it will die, Cain and Roughgarden (?) took a different mathemathical approach. Recognizing that Cohen's "bang/bang" allocation schedule only works for a season of a given length, they argued that a plant might well allocate some resources to reproduction early on. It will produce less then than later, but with a graded reproductive schedule it can hedge its bets. If a season is random (as all seasons are), then a plant could easily wait too long to reproduce and lose altogether the possibility of doing so.
Cain and Roughgarden's model was as extreme as Cohen's, at the other end of the spectrum: they devised a model in which every day is equally likely to be the day that the plant dies. But no one has done the math of an arbitrary probability distribution. Couldn't something like actuarial tables be constructed, looking at several decades of rainfall and frost (for instance), and then building statistics into the model that reflect how likely a plant is to die at any given point in the season? But the only patterns available are equally simple ones. Models in general represent trade-offs between numerical precision, generability, and mechanistic realism (or explanatory power). Cohen's model, for instance, which deals only with a situation of perfect knowledge, has very specific applicability, but is not generalizable, and--given different efficiencies of age, different lengths of growing seasons--not numerically precise.
Ted ended his presentation by describing what he himself has done with Cohen's model: applying an evolutionary algorithm to the schedule to maximize seed mass in different environmental situations. Looking for the schedule with the highest fitness, he selects random candidate solutions, lets them reproduce for 2,000 generations, picks the best (erasing all the rest), varies them slightly, then multiplies them for another 2,000 generations of exposure to the environment. Fitness (as exemplified in the graphs) evolves in these populations following (surprise!) the "bang/bang" allocation schedule.
The further discussion of such questions is invited to continue in the on-line forum. The semester's series will continue next Friday, Feb. 18, when Hiroshi Iwasaki of the Theater Program will talk about "Who's Watching?-- Imaginary Dialogues."